Global thermonuclear war would destroy technological civilization, but even now there are small groups of pastoralists and hunter-gatherers in remote regions who use only muscle power and do not need factory manufactured goods. Uncontacted tribes [1] would go on living much as usual if a nuclear war depopulated industrialized regions.

There would be some increase in mortality among remote tribes as radioactive fallout from a large scale nuclear war drifted down globally, but that wouldn't be enough to cause human extinction.

[1] https://en.wikipedia.org/wiki/Uncontacted_peoples